Internal problem ID [10499]
Internal file name [OUTPUT/9447_Monday_June_06_2022_02_32_58_PM_31295989/index.tex]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-1. Equations with
sine
Problem number: 2.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "riccati"
Maple gives the following as the ode type
[_Riccati]
\[ \boxed {y^{\prime }-y^{2}=-a^{2}+a \lambda \sin \left (\lambda x \right )+a^{2} \sin \left (\lambda x \right )^{2}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= y^{2}-a^{2}+a \lambda \sin \left (\lambda x \right )+a^{2} \sin \left (\lambda x \right )^{2} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = y^{2}-a^{2}+a \lambda \sin \left (\lambda x \right )+a^{2} \sin \left (\lambda x \right )^{2} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-a^{2}+a \lambda \sin \left (\lambda x \right )+a^{2} \sin \left (\lambda x \right )^{2}\), \(f_1(x)=0\) and \(f_2(x)=1\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{u} \tag {1} \end {align*}
Using the above substitution in the given ODE results (after some simplification)in a second order ODE to solve for \(u(x)\) which is \begin {align*} f_2 u''(x) -\left ( f_2' + f_1 f_2 \right ) u'(x) + f_2^2 f_0 u(x) &= 0 \tag {2} \end {align*}
But \begin {align*} f_2' &=0\\ f_1 f_2 &=0\\ f_2^2 f_0 &=-a^{2}+a \lambda \sin \left (\lambda x \right )+a^{2} \sin \left (\lambda x \right )^{2} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} u^{\prime \prime }\left (x \right )+\left (-a^{2}+a \lambda \sin \left (\lambda x \right )+a^{2} \sin \left (\lambda x \right )^{2}\right ) u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = {\mathrm e}^{\frac {\sin \left (\lambda x \right ) a}{\lambda }} \left (c_{1} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )+c_{2} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = {\mathrm e}^{\frac {\sin \left (\lambda x \right ) a}{\lambda }} \left (c_{2} \left (\cos \left (\lambda x \right ) \sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) a +\frac {\lambda \cos \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )}{2}\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )+\cos \left (\lambda x \right ) \left (\frac {c_{2} \lambda \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )}{2}+\operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right ) c_{1} a +\frac {c_{1} \lambda \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )}{2}\right )\right ) \] Using the above in (1) gives the solution \[ y = -\frac {c_{2} \left (\cos \left (\lambda x \right ) \sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) a +\frac {\lambda \cos \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )}{2}\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )+\cos \left (\lambda x \right ) \left (\frac {c_{2} \lambda \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )}{2}+\operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right ) c_{1} a +\frac {c_{1} \lambda \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )}{2}\right )}{c_{1} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )+c_{2} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {-\left (\cos \left (\lambda x \right ) \sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) a +\frac {\lambda \cos \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )}{2}\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )-\cos \left (\lambda x \right ) \left (\frac {\lambda \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )}{2}+\operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right ) c_{3} a +\frac {c_{3} \lambda \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )}{2}\right )}{c_{3} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )+\sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {-\left (\cos \left (\lambda x \right ) \sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) a +\frac {\lambda \cos \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )}{2}\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )-\cos \left (\lambda x \right ) \left (\frac {\lambda \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )}{2}+\operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right ) c_{3} a +\frac {c_{3} \lambda \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )}{2}\right )}{c_{3} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )+\sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {-\left (\cos \left (\lambda x \right ) \sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) a +\frac {\lambda \cos \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )}{2}\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )-\cos \left (\lambda x \right ) \left (\frac {\lambda \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )}{2}+\operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right ) c_{3} a +\frac {c_{3} \lambda \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )}{2}\right )}{c_{3} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )+\sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{2}=-a^{2}+a \lambda \sin \left (\lambda x \right )+a^{2} \sin \left (\lambda x \right )^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{2}-a^{2}+a \lambda \sin \left (\lambda x \right )+a^{2} \sin \left (\lambda x \right )^{2} \end {array} \]
Maple trace Kovacic algorithm successful
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying Riccati trying Riccati Special trying Riccati sub-methods: trying Riccati_symmetries trying Riccati to 2nd Order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = (a^2-a*lambda*sin(lambda*x)-a^2*sin(lambda*x)^2)*y(x), y(x)` *** Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying an equivalence, under non-integer power transformations, to LODEs admitting Liouvillian solutions. -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Reducible group (found an exponential solution) Group is reducible, not completely reducible Solution has integrals. Trying a special function solution free of integrals... -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Whittaker -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius <- Heun successful: received ODE is equivalent to the HeunC ODE, case a <> 0, e <> 0, c = 0 <- Kovacics algorithm successful <- Equivalence, under non-integer power transformations successful Change of variables used: [x = arccos(t)/lambda] Linear ODE actually solved: (2*a*lambda*(-t^2+1)^(1/2)-2*a^2*t^2)*u(t)-2*lambda^2*t*diff(u(t),t)+(-2*lambda^2*t^2+2*lambda^2)*diff(diff(u(t),t),t) = <- change of variables successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 289
dsolve(diff(y(x),x)=y(x)^2-a^2+a*lambda*sin(lambda*x)+a^2*sin(lambda*x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-2 a c_{1} \cos \left (x \lambda \right ) \sin \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right )-c_{1} \lambda \cos \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right )\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (x \lambda \right )}{2}+\frac {1}{2}\right )-2 \left (a \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (x \lambda \right )}{2}+\frac {1}{2}\right )+\frac {\lambda \left (\operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (x \lambda \right )}{2}+\frac {1}{2}\right ) c_{1} \sin \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right )+\operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (x \lambda \right )}{2}+\frac {1}{2}\right )\right )}{2}\right ) \cos \left (x \lambda \right )}{2 \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (x \lambda \right )}{2}+\frac {1}{2}\right ) \sin \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right ) c_{1} +2 \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (x \lambda \right )}{2}+\frac {1}{2}\right )} \]
✓ Solution by Mathematica
Time used: 4.337 (sec). Leaf size: 132
DSolve[y'[x]==y[x]^2-a^2+a*\[Lambda]*Sin[\[Lambda]*x]+a^2*Sin[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {a c_1 \cos (\lambda x) \int _1^xe^{-\frac {2 a \sin (\lambda K[1])}{\lambda }}dK[1]+a \cos (\lambda x)+c_1 e^{-\frac {2 a \sin (\lambda x)}{\lambda }}}{1+c_1 \int _1^xe^{-\frac {2 a \sin (\lambda K[1])}{\lambda }}dK[1]} \\ y(x)\to -\frac {e^{-\frac {2 a \sin (\lambda x)}{\lambda }}}{\int _1^xe^{-\frac {2 a \sin (\lambda K[1])}{\lambda }}dK[1]}-a \cos (\lambda x) \\ \end{align*}